2 L
. Smith Mathematical Social Sciences 38 1999 1 –9
weakly more than a reservation wage, equal to his expected value from further search. Think of unemployment as a financial security, affording the right to search further, and
secure any wage found. In a stationary world, it is well-known that the reservation wage is the net flow return to this asset.
In this decision theory paper, I venture that the known wage distribution evolves deterministically. While this assumption is best seen as a stepping stone to a more
general analysis, it aptly captures the temporary employment dynamics of Smith 1995. In the standard steady-state theory, quits do not occur, for if a job is optimal to accept, it
is optimal to retain forever. If the worker is indentured to his job, the above financial intuition obtains. Instead I assume that the worker may retain any position at his
pleasure, quitting at will. Optimal exercise of this quit option adds a nice twist to the standard analysis. I consider the resulting link between the reservation wage and the
unemployment value, underscoring that only a non-local relationship exists. I then observe that the worker’s reservation wage becomes the flow return on the unemploy-
ment ‘asset’ with an option to renew at the same rate.
The problem can be more generally cast as an individual searching for many- dimensional ‘prizes’, rather than jobs. While this does not enrich the above analysis, a
continuity result is best understood in this setting. I show that the optimal acceptance set is a lower hemicontinuous correspondence of time. As a result, the reservation wage is
necessarily an upper semi-continuous function of time. The paper concludes with a simple example of how some well-known comparative statics may fail in a nonstationary
environment, and discuss some new ones that arise.
2. The model
Consider a continuous time setting, where an individual either has a job, or is searching for one while paying a flow search cost c. The income flow I at any time t is
t
thus either 2 c or some positive wage, generically denoted w. The word ‘cost’ is loosely applied here, since c 0 or c , 0 is possible: In a macroeconomic context, 2 c might
well measure unemployment compensation. Even c 5 0 is permissible, in which case there is a pure time cost of search, owing to worker impatience: He discounts future
payoffs at the interest rate r . 0.
Job arrivals follow a Poisson process with parameter r . 0. Thus, an offer arrives
with chance about rdt in a small dt time span. The distribution of wage offers that are
tendered deterministically evolves with time, with the c.d.f. process hF wut, w 0j. For
t
each t, the distribution F has support on [0,` with bounded finite mean; moreover,
t
t∞F w is a measurable function of time for every wage w. For instance, the wage
t
distribution may be fixed, or shift finitely or countably many times, or perhaps continuously with the passage of time.
For simplicity, the individual cannot search while working, and so cannot hold more than one job at a time. This reflects the opportunity cost inherent in accepting a job—a
tradeoff at the heart of wage search models. Otherwise, the problem is trivial: The first positive wage is accepted, and thereafter any better-paying offer. More generally, a
tradeoff arises so long as there is a lower flow arrival rate of offers while working than
L . Smith Mathematical Social Sciences 38 1999 1 –9
3
searching. Any job may be retained forever if desired, or freely dropped at will. Search costs are assumed low enough that one will never stop searching altogether, but not so
low and negative as to render working unprofitable. The individual chooses which wages to accept while searching, and ongoing jobs to
quit, to maximize the expected present value of future wages less search costs borne. Since the costs and outside options of current and proposed jobs are identical, they must
be treated symmetrically. Further, the deterministic evolution of kF l implies that an
t
optimal strategy is open loop known at time 0 and not in feedback form: It cannot depend on the current wage or wage history, for instance. Thus, the individual accepts or
retains any wage w belonging to an acceptance set at time t. A strategy is then a
t
measurable time-path of acceptance sets k R1ut0l, which is a real correspondence
t
of time. Given this strategy, his supremum average value is then
` 2rs 2t
V 5 sup
E
re I ds
1
t kA ,stl
s
s
t
expecting over the income process kI l wage arrival times and realizations.
s
3. The main results
